arXiv:1608.07084v2 [math.PR] 2 Sep 2016
Strengthened volume inequalities for L p zonoids of even isotropic measures ∗
K´aroly J. B¨or¨oczky, Ferenc Fodor, Daniel Hug September 24, 2018
Abstract
We strengthen the volume inequalities for Lp zonoids of even isotropic measures and for their duals, which are due to Ball, Barthe and Lutwak, Yang, Zhang. Along the way, we prove a stronger version of the Brascamp-Lieb inequality for a family of functions that can approximate arbitrary well some Gaussians when equality holds. The special casep =
∞yields a stability version of the reverse isoperimetric inequality for centrally symmetric bodies.
1 Introduction
According to the classical isoperimetric inequality Euclidean balls minimize the surface area among convex bodies of given volume in Euclidean spaceRn. We call a subset of Rna convex body if it is compact, convex and has non-empty interior. Let Bn be the Euclidean unit ball centred at the origin, and letS(·)andV(·)denote the surface area and the volume functional in Rn, respectively. The isoperimetric inequality can be stated in the form
S(Bn)n
V(Bn)n−1 ≤ S(K)n V(K)n−1,
where equality holds if and only if K is a Euclidean ball. Recently, N. Fusco, F. Maggi, A.
Pratelli [25] proved an essentially optimal stability version of the isoperimetric inequality. It states that ifK is a convex body withV(K) = V(Bn)and ifS(Bn) ≥ (1−ε)S(K)holds for some smallε >0, thenKis close to some translateBn+x,x∈Rn, of the unit ball; namely,
V(K∆(Bn+x))≤γε1/2,
whereγ >0depends only onn, and∆denotes the symmetric difference of sets.
∗AMS 2010 subject classification. Primary 52A40; Secondary 52A38, 52B12, 26D15.
Key words and phrases. Surface area, volume, isoperimetric inequality, reverse isoperimetric inequality, John ellip- soid, parallelotope,Lp-zonoid, Brascamp-Lieb inequality, mass transportation, stability result, isotropic measure.
Stability estimates for the planar isoperimetric inequality go back to the works of Minkowski and Bonnesen. However, a systematic exploration is much more recent. We refer to the sur- vey articles of H. Groemer [27, 28] for an introduction to geometric stability results. The recent monograph [46] by R. Schneider provides an up-to-date treatment of the topic including ap- plications. Here we only note that the stability estimate related to the isoperimetric inequality obtained in [25] was extended to a stability version of the Brunn-Minkowski inequality by A. Fi- galli, F. Maggi, A. Pratelli [23, 24].
Aiming at a reverse isoperimetric inequality, F. Behrend [10] suggested to consider equiva- lence classes of convex bodies with respect to non-singular linear transformations. C.M. Petty [45] proved (see also A. Giannopoulos, M. Papadimitrakis [26]) that there is an essentially unique representative minimizing the isoperimetric ratio in each equivalence class. The unique mini- mizer in an equivalence class is characterized by the property that its suitably normalized area measure is isotropic. We give a precise definition of isotropic measures later. This characteri- zation yields that cubes minimize the isoperimetric ratio within the class of parallelotopes, and regular simplices within the class of simplices.
The functional that assigns to each equivalence class the minimum of the isoperimetric ra- tio within that class is affine invariant and upper semi-continuous, therefore it attains its max- imum on the affine equivalence classes of convex bodies. In the Euclidean plane, the method of F. Behrend [10] yields that the maximum is attained by the affine equivalence class of tri- angles, and by the affine equivalence class of parallelograms if the convex body is assumed to be centrally symmetric. The extension of these results to higher dimensions proved to be quite difficult. Decades after Behrend’s paper, K.M. Ball in [1, 3] managed to establish reverse forms of the isoperimetric inequality in arbitrary dimensions. More precisely, the largest isoperimetric ratio is attained by simplices according to [3], and by parallelotopes among centrally symmetric convex bodies according to [1]. Since the reverse isoperimetric inequality and a stronger form of it for general convex bodies are discussed in K.J. B¨or¨oczky, D. Hug [13], in this paper we concentrate on centrally symmetric convex bodies.
In order to state the result of K.M. Ball [1] about centrally symmetric convex bodies, we set Wn= [−1,1]n, and note thatS(Wn) =n2n=nV(Wn).
Theorem A (K.M. Ball) For any centrally symmetric convex body K in Rn, there exists some Φ∈GL(n)such that
S(ΦK)n
V(ΦK)n−1 ≤ S(Wn)n
V(Wn)n−1. (1)
The case of equality in Theorem A was settled by F. Barthe [6]. He proved that if the left side of (1) is minimized over allΦ∈GL(n), then equality holds precisely whenKis a parallelotope.
Our first objective is to prove a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Following [23–25], we define an affine invariant distance of origin symmetric convex bodiesKandM based on the volume difference. Letα =V(K)−1/n, β =V(M)−1/n, and define
δvol(K, M) = min{V (Φ(αK)∆(βM)) : Φ ∈SL(n)}
whereSL(n) is the group of linear transformations ofRn of determinant one. In fact, δvol(·,·) induces a metric on the linear equivalence classes of origin symmetric convex bodies.
The John ellipsoid of a convex body K in Rn is the unique maximum volume ellipsoid contained inK. IfK is origin symmetric, then its John ellipsoid is also origin symmetric. Note that each convex body has an affine image whose John ellipsoid is Bn. The John ellipsoid is a frequently used tool in geometric analysis, and, in particular, it was used by K.M. Ball in the proof of the reverse isoperimetric inequality. Since we will use the John ellipsoid in our arguments, below we review its basic properties (see (2)). For a more detailed treatment of the topic, we refer to K.M. Ball [4], P.M. Gruber [30] and R. Schneider [46].
Theorem 1.1 LetK be an origin symmetric convex body inRn, n ≥3, whose John ellipsoid is a Euclidean ball, and letε ∈[0,1). Ifδvol(K, Wn)≥ε, then
S(K)n
V(K)n−1 ≤(1−γ ε3) S(Wn)n V(Wn)n−1, whereγ =n−cn3 for some absolute constantc >0.
The stability order (the exponent 3 of ε) in Theorem 1.1 is close to be optimal, but most probably it is not optimal. Considering a convex bodyKwhich is obtained fromWnby cutting off simplices of heightεat the vertices ofWn, one can see that the exponent ofεmust be at least 1in Theorem 1.1.
Another common affine invariant distance between convex bodies is the Banach-Mazur met- ricδBM(K, M), which we define here only for origin symmetric convex bodiesK andM. Let
δBM(K, M) = log min{λ≥1 : K ⊆Φ(M)⊆λ K for someΦ∈GL(n)}.
We note thatδvol ≤ 2n2δBM(see, say, [13]). Furthermore, δBM ≤ γ δvoln1 , where γ depends only on the dimensionn(see [12, Section 5]). The example of a ball from which a cap is cut off shows that in the latter inequality the exponent n1 cannot be replaced by anything larger than n+12 . Theorem 1.2 LetK be an origin symmetric convex body inRn, n ≥3, whose John ellipsoid is a Euclidean ball, and letε ∈[0,1). IfδBM(K, Wn)≥ε, then
S(K)n
V(K)n−1 ≤(1−γ εn) S(Wn)n V(Wn)n−1, whereγ =n−cn3 for some absolute constantc >0.
The stability order (the exponentnofε) in Theorem 1.2 is again close to be optimal, but very likely it is not optimal. Considering a convex bodyK which is obtained fromWnby cutting off simplices of heightε at the vertices of Wn, one can see that the exponent ofε must be at least n−1in Theorem 1.2.
In the planar case, a modification of the argument of F. Behrend [10] leads to stability results of optimal order.
Theorem 1.3 Let K be an origin symmetric convex body in R2 which has a square as an in- scribed parallelogram of maximum area. Letε∈[0,1). Ifδvol(K, W2)≥εorδBM(K, W2)≥ε,
then S(K)2
V(K) ≤ 1− ε
54
S(W2)2 V(W2).
Note that for an origin symmetric convex body K inR2 there always exists a linear trans- formΦ ∈ GL(2) such that a square is an inscribed parallelogram of maximum area of ΦK. In particular, if we define ir(K) = min{S(ΦK)2/V(ΦK) : Φ ∈GL(2)}, for an origin symmetric convex body inKinR2, and ifε∈[0,1), then Theorem 1.3 implies that
ir(K)≤ 1− ε
54
ir(W2) provided thatδvol(K, W2)≥εorδBM(K, W2)≥ε.
As mentioned before, the proof of the reverse isoperimetric inequality by K.M. Ball [1, 3]
is based on a volume estimate for convex bodies whose John ellipsoid is the unit ball Bn. Let Sn−1denote the Euclidean unit sphere. According to a classical theorem of F. John [33] (see also K.M. Ball [4]),Bnis the ellipsoid of maximal volume in an origin symmetric convex bodyK if and only ifBn ⊆K and there exist±u1, . . . ,±uk∈Sn−1∩∂K andc1, . . . , ck>0such that
Xk
i=1
ciui⊗ui = Idn, (2)
where⊗denotes the tensor product of vectors inRn, Idndenotes then×nidentity matrix and
∂K is the boundary ofK.
Following A. Giannopoulos, M. Papadimitrakis [26] and E. Lutwak, D. Yang, G. Zhang [42], we call an even Borel measureµon the unit sphereSn−1isotropic if
Z
Sn−1
u⊗u dµ(u) = Idn.
In this case, equating traces of both sides we obtain thatµ(Sn−1) =n.
Using the standard notation h·,·ifor the Euclidean scalar product and k · kfor the induced norm inRn, the support functionhK of a convex compact setKinRnatv ∈Rnis defined as
hK(v) = max{hv, xi: x∈K}.
For any p ≥ 1and an even measure µon Sn−1 not concentrated on any great subsphere, we define theLpzonoidZp(µ)associated withµby
hZp(µ)(v)p = Z
Sn−1|hu, vi|pdµ(u), which is a zonoid in the classical sense ifp= 1. In addition, let
Z∞(µ) = lim
p→∞Zp(µ) = conv suppµ,
and for1≤p≤ ∞, letZp∗(µ)be the polar ofZp(µ). In particular, Zp∗(µ) =
x∈Rn : Z
Sn−1|hx, ui|pdµ(u)≤1
forp∈[1,∞), Z∞∗ (µ) = {x∈Rn: hx, ui ≤1foru∈suppµ},
and henceZ2(µ) = Bnfor any even isotropic measureµ.
It follows from D.R. Lewis [37] (see also E. Lutwak, D. Yang and G. Zhang [40, 41]) that any n-dimensional subspace of Lp is isometric tok · kZp∗(µ) for some isotropic measureµonSn−1, where
kxkZp∗(µ) = Z
Sn−1|hx, ui|pdµ(u) 1p
, x∈Rn.
We call a measure ν onSn−1 a cross measure if there is an orthonormal basis u1, . . . , un of Rnsuch that
suppν ={±u1, . . . ,±un},
andν({ui}) = ν({−ui}) = 1/2fori = 1, . . . , n, and hence ν is even and isotropic. We fix a cross measureνnonSn−1. We note that ifp∈[1,∞], andΓ(·)is Euler’s Gamma function, then
V(Zp(νn)) =
Γ(1+n2)Γ(1+p2)
Γ(1+12)Γ(1+n+p2 ) ifp≥1,
2n
n! ifp=∞.
In addition,
V(Zp∗(νn)) =
2nΓ(1+
1 p)n
Γ(1+np) ifp≥1, 2n ifp=∞.
The crucial statement leading to the reverse isoperimetric inequality is the case ofZ∞∗ (µ).
Theorem B Ifµis an even isotropic measure onSn−1 andp∈[1,∞], then V(Zp(µ)) ≥ V(Zp(νn)),
V(Zp∗(µ)) ≤ V(Zp∗(νn)).
Assumingp6= 2, equality holds if and only ifµis a cross measure.
Theorem B is the work of K.M. Ball [3] and F. Barthe [6] ifµis discrete, and their method was extended to arbitrary even isotropic measuresµby E. Lutwak, D. Yang, and G. Zhang [40].
The measures onSn−1which have an isotropic linear image are characterized by K.J. B¨or¨oczky, E. Lutwak, D. Yang and G. Zhang [14], building on the works of E.A. Carlen, and D. Cordero- Erausquin [17], J. Bennett, A. Carbery, M. Christ and T. Tao [11] and B. Klartag [36]. We note that isotropic measures onRnplay a central role in the KLS conjecture by R. Kannan, L. Lov´asz and M. Simonovits [34]; see, for instance, F. Barthe and D. Cordero-Erausquin [8], O. Guedon and E. Milman [32] and B. Klartag [35].
To state a stability version of Theorem B, a natural notion of distance between two isotropic measures µ and ν is the Wasserstein distance (also called the Kantorovich-Monge-Rubinstein distance) δW(µ, ν). To define it, we write ∠(v, w)to denote the angle between non-zero vec- torsv andw; that is, the geodesic distance of the unit vectors kvk−1v and kwk−1w on the unit sphere. LetLip1(Sn−1)denote the family of Lipschitz functions with Lipschitz constant at most 1; namely,f : Sn−1 → Ris inLip1(Sn−1)ifkf(x)−f(y)k ≤∠(x, y)forx, y ∈ Sn−1. Then the Wasserstein distance ofµandνis given by
δW(µ, ν) = max Z
Sn−1
f dµ− Z
Sn−1
f dν: f ∈Lip1(Sn−1)
.
What we actually need in this paper is the Wasserstein distance of an isotropic measureµfrom the closest cross measure. Therefore, in the case of two isotropic measuresµandν, we define
δWO(µ, ν) = min{δW(µ,Φ∗ν) : Φ∈O(n)} whereΦ∗νdenotes the pushforward ofν byΦ :Sn−1 →Sn−1.
Theorem 1.4 Let µbe an even isotropic measure on Sn−1, n ≥ 2, let ε ∈ [0,1), and letp ∈ [1,∞]withp6= 2. IfδWO(µ, νn)≥ε >0, then
V(Zp(µ)) ≥ (1 +γε3)V(Zp(νn)), V(Zp∗(µ)) ≤ (1−γε3)V(Zp∗(νn)) whereγ =n−cn3min{|p−2|2,1}for an absolute constantc >0.
To state another stability version of Theorem B, in the casep = ∞, we use the “spherical”
Hausdorff distanceδH(X, Y)of compact setsX, Y ⊆Sn−1 given by δH(X, Y) = min
maxx∈X min
y∈Y ∠(x, y),max
y∈Y min
x∈X∠(x, y)
. In addition, let
δHO(X, Y) = min{δH(X,ΦY) : Φ∈O(n)}.
We note that if δHO(suppµ,suppνn) ≤ 1/(7n2) for an even isotropic measure µ, then δW O(µ, νn) ≤ 2nδHO(suppµ,suppνn)according to Corollary 6.2. However, as we will see in Section 9, Theorem 1.4 implies the following seemingly stronger statement in the casep=∞. Corollary 1.5 Ifµis an even isotropic measure onSn−1, andδHO(suppµ,suppνn) ≥ ε > 0, then
V(Z∞(µ)) ≥ (1 +γε3)V(Z∞(νn)), V(Z∞∗ (µ)) ≤ (1−γε3)V(Z∞∗ (νn)) whereγ =n−cn3 for an absolute constantc >0.
We note that the order ε3 of the error term in Corollary 1.5 can be improved to ε if n = 2 according to Theorem 11.1.
The proof of Theorem B by is based on the rank one case of the geometric Brascamp-Lieb inequality. An essential tool in our approach is the proof provided by F. Barthe [5, 6], which is based on mass transportation. Therefore, we review the argument from [5] in Section 2. At the end of that section, we outline the arguments leading to Theorem 1.1, Theorem 1.2 and Theorem 1.4 and we describe the structure of the paper. We also indicate in Section 2 what stability result can be expected concerning the Brascamp-Lieb inequality (see Conjecture 2.1).
Along the way of proving our main statements, we also establish some properties of arbitrary (not only even) isotropic measures in Section 5 that might be useful in other applications as well.
Let us point out that the corresponding question in the non-symmetric setting is wide open.
We call an isotropic measureµonSn−1 centred if Z
Sn−1
u dµ(u) =o.
Here and in the following, we write o for the origin (the zero vector). For a centred isotropic measureµonSn−1, and forp∈[1,∞), we define the non-symmetricLp zonoidZp(µ)by
hZp(µ)(v)p = 2 Z
Sn−1
max{0,hv, ui}pdµ(u), Zp∗(µ) =
x∈Rn: Z
Sn−1
max{0,hx, ui}pdµ(u)≤ 1 2
.
This notion (for any discrete measure onSn−1, not only isotropic ones), occurs in M. Webern- dorfer [47] in connection with reverse versions of the Blaschke-Santal´o inequality. The fac- tor 2 is included to match the earlier definition for even isotropic measures. The difference to the case of even isotropic measures is that if p = 2 and µis a non-even centered isotropic measure, then Z2(µ)is typically not a Euclidean ball but has constant squared width; namely, hZp(µ)(v)2+hZp(µ)(−v)2 is constant forv ∈Sn−1.
Conjecture 1.6 Ifµis a centered isotropic measure onSn−1 andp ∈[1,∞), moreoverν is an isotropic measure onSn−1such thatsuppνconsists of the vertices of a regular simplex, then
V(Zp(µ)) ≥ V(Zp(ν)), V(Zp∗(µ)) ≤ V(Zp∗(ν)).
If µis a centered isotropic measure on Sn−1, then Z∞(µ) = conv suppµ. In particular, if p = ∞, then (3) was proved by K.M. Ball in [3] for discreteµ, (3) was proved by F. Barthe in [6] again for discreteµ, and the case of general centered isotropicµwas handled E. Lutwak, D.
Yang and G. Zhang [42].
An inequality related to the case p = 2of Conjecture 1.6 is proved by E. Lutwak, D. Yang, G. Zhang [43].
2 A brief review of the Brascamp-Lieb and the reverse Brascamp-Lieb inequality
The rank one geometric Brascamp-Lieb inequality (3), identified by K.M. Ball [1] as an essential case of the rank one Brascamp-Lieb inequality, due to H.J. Brascamp, E.H. Lieb [15], and the reverse form (4), due to F. Barthe [5, 6], read as follows. Ifu1, . . . , uk ∈ Sn−1 are distinct unit vectors andc1, . . . , ck >0satisfy
Xk
i=1
ciui⊗ui = Idn, andf1, . . . , fkare non-negative measurable functions onR, then
Z
Rn
Yk
i=1
fi(hx, uii)cidx ≤ Yk
i=1
Z
R
fi ci
, and (3)
Z ∗
Rn
sup
x=Pk i=1ciθiui
Yk
i=1
fi(θi)cidx ≥ Yk
i=1
Z
R
fi
ci
. (4)
In (4), the supremum extends over all θ1, . . . , θk ∈ R. Since the integrand need not be a mea- surable function, we have to consider the outer integral. If k = n, then u1, . . . , un form an orthonormal basis and thereforeθ1, . . . , θkare uniquely determined for a givenx∈Rn.
According to F. Barthe [6], if equality holds in (3) or in (4) and none of the functions fi
is identically zero or a scaled version of a Gaussian, then there is an origin symmetric regular crosspolytope inRnsuch thatu1, . . . , uklie among its vertices. Conversely, equality holds in (3) and (4) if each fi is a scaled version of the same centered Gaussian, or ifk =n andu1, . . . , un
form an orthonormal basis.
A thorough discussion of the rank one Brascamp-Lieb inequality can be found in E. Carlen, D. Cordero-Erausquin [17]. The higher rank case, due to E.H. Lieb [38], is reproved and further explored by F. Barthe [6] (including a discussion of the equality case), and is again carefully anal- ysed by J. Bennett, T. Carbery, M. Christ, T. Tao [11]. In particular, see F. Barthe, D. Cordero- Erausquin, M. Ledoux, B. Maurey [9] for an enlightening review of the relevant literature and an approach via Markov semigroups in a quite general framework.
F. Barthe [5, 6] provided concise proofs of (3) and (4) based on mass transportation (see also K.M. Ball [4] for (3)). We sketch the main ideas of his approach, since it will be the starting point of subsequent refinements.
We assume that eachfi is a positive continuous probability density both for (3) and (4), and letg(t) = e−πt2 be the Gaussian density. Fori = 1, . . . , k, we consider the transportation map Ti :R→Rsatisfying
Z t
−∞
fi(s)ds= Z Ti(t)
−∞
g(s)ds.
It is easy to see thatTiis bijective, differentiable and
fi(t) =g(Ti(t))·Ti′(t), t∈R. (5)
To these transportation maps, we associate the smooth transformationΘ :Rn →Rngiven by Θ(x) =
Xk
i=1
ciTi(hui, xi)ui, x∈Rn, which satisfies
dΘ(x) = Xk
i=1
ciTi′(hui, xi)ui⊗ui.
In this case, dΘ(x)is positive definite and Θ :Rn → Rn is injective (see [5, 6]). We will need the following two estimates due to K.M. Ball [1] (see also [6] for a simpler proof of (i)).
(i) For anyt1, . . . , tk >0, we have det
Xk
i=1
ticiui⊗ui
!
≥ Yk
i=1
tcii.
(ii) Ifz =Pk
i=1ciθiui forθ1, . . . , θk ∈R, then kzk2 ≤
Xk
i=1
ciθ2i. (6)
Therefore, using first (5), then (i) withti = Ti′(hui, xi), the definition ofΘand (ii), and finally the transformation formula, the following argument leads to the Brascamp-Lieb inequality (3).
Z
Rn
Yk
i=1
fi(hui, xi)cidx= Z
Rn
Yk
i=1
g(Ti(hui, xi))ci
! k Y
i=1
Ti′(hui, xi)ci
!
dx (7)
≤ Z
Rn
Yk
i=1
e−πciTi(hui,xi)2
! det
Xk
i=1
ciTi′(hui, xi)ui⊗ui
!
dx (8)
≤ Z
Rn
e−πkΘ(x)k2det (dΘ(x))dx
≤ Z
Rn
e−πkyk2dy = 1.
The Brascamp-Lieb inequality (3) for arbitrary non-negative integrable functions fi follows by scaling and approximation.
For the reverse Brascamp-Lieb inequality (4), we consider the inverseSiofTi, and hence Z t
−∞
g(s)ds= Z Si(t)
−∞
fi(s)ds,
g(t) = fi(Si(t))·Si′(t), t∈R. (9) In addition,
dΨ(x) = Xk
i=1
ciSi′(hui, xi)ui⊗ui holds for the smooth transformationΨ :Rn→Rngiven by
Ψ(x) = Xk
i=1
ciSi(hui, xi)ui, x∈Rn.
In particular,dΨ(x)is positive definite andΨ : Rn → Rnis injective (see [5, 6]). Therefore (i) and (9) lead to
Z ∗
Rn
sup
x=Pk i=1ciθiui
Yk
i=1
fi(θi)cidx
≥ Z ∗
Rn
sup
Ψ(y)=Pk i=1ciθiui
Yk
i=1
fi(θi)ci
!
det (dΨ(y))dy
≥ Z
Rn
Yk
i=1
fi(Si(hui, yi))ci
! det
Xk
i=1
ciSi′(hui, yi)ui⊗ui
!
dy (10)
≥ Z
Rn
Yk
i=1
fi(Si(hui, yi))ci
! k Y
i=1
Si′(hui, yi)ci
!
dy (11)
= Z
Rn
Yk
i=1
g(hui, yi)ci
! dy=
Z
Rn
e−πkyk2dy= 1.
Again, the reverse Brascamp-Lieb inequality (4) for arbitrary non-negative integrable functions fifollows by scaling and approximation.
We observe that (i) shows that the optimal constant in the geometric Brascamp-Lieb inequal- ity is1. The stability version of (i) (withvi =√ciui), Lemma 3.1, is an essential tool in proving a stability version of the Brascamp-Lieb inequality leading to Theorem 1.4.
Even if we do not use it in this paper, we point out that F. Barthe [7] proved “continuous”
versions of the Brascamp-Lieb and the reverse Brascamp-Lieb inequalities that work for any isotropic measure µ on Sn−1 (see (12) and (13) below). Here we only consider the case in which all non-negative real functions involved coincide with a “nice” probability density func- tion, which is the common case in geometric applications. So letf : R → [0,∞)be such that R
Rf = 1andsupp(f) = [a, b]for somea, b∈ [−∞,∞]. Further, we assume thatf is positive and continuous on[a, b]. According to [7], we have
Z
Rn
exp Z
Sn−1
logf(hx, ui)dµ(u)
dx≤ 1. (12)
For the reverse inequality, leth:Rn→[0,∞)be a measurable function which satisfies h
Z
Sn−1
θ(u)u dµ(u)
≥exp Z
Sn−1
logf(θ(u))dµ(u)
for any continuous functionθ : suppµ→R. Then, we have Z
Rn
h≥1. (13)
Let us briefly discuss how K.M. Ball [1] and F. Barthe [6] used the Brascamp-Lieb inequality and its reverse form to prove the discrete version of Theorem B. In this section, we writeµto denote the isotropic measure onSn−1 whose support is{u1, . . . , uk}withµ({ui}) =ci, and we assume thatµis an even measure. For i= 1, . . . , k, we consider the probability densities onR (see (19)) given by
fi(t) = 1
2Γ(1 + 1p)e−|t|p, t ∈R, ifp∈[1,∞), andfi = 121[−1,1]ifp=∞, where
1[−1,1](t) =
1 ift∈[−1,1], 0 otherwise.
We will frequently use the following observation due to K. Ball [3]. IfK is an orgin symmetric convex body inRnwith associated normk · kK and ifp∈[1,∞), then
V(K) = 1 Γ(1 + np)
Z
Rn
e−kxkpK dx, where
kxkK = min{λ ≥0 : x∈λK}, x∈Rn. In particular, ifp∈[1,∞), then
V(Zp∗(µ)) = 1 Γ(1 + np)
Z
Rn
exp − Xk
i=1
ci|hx, uii|p
! dx
= 2nΓ
1 + 1pn
Γ(1 + np) Z
Rn
Yk
i=1
fi(hx, uii)cidx (14)
≤
2nΓ
1 + 1pn
Γ(1 + np) Yk
i=1
Z
R
fi
ci
= 2nΓ
1 + 1pn
Γ(1 + np) . (15) On the other hand, ifp=∞, then usingfi = 121[−1,1], we have
V(Z∞∗ (µ)) = 2n Z
Rn
Yk
i=1
fi(hx, uii)cidx≤2n Yk
i=1
Z
R
fi ci
= 2n.
Equality in (15) leads to equality in the Brascamp-Lieb inequality, and hence k = 2n and u1, . . . , ukform the vertices of a regular crosspolytope inRn.
For the lower bound on the volume of the Lp zonotopes andp ∈ [1,∞], let us choosep∗ ∈ [1,∞]such that 1p + p1∗ = 1. Ifp∈[1,∞), then an (auxiliary) origin symmetric convex body is defined by
Mp(µ) = ( k
X
i=1
ciθiui : Xk
i=1
ci|θi|p ≤1 )
.
We drop the reference toµ, if it does not cause any misunderstanding. In particular, kxkMp = inf
x=Pk i=1ciθiui
Xk
i=1
ci|θi|p
!1p
, x∈Rn.
In addition, we define
M∞(µ) = ( k
X
i=1
ciθiui : |θi| ≤1fori= 1, . . . , k )
. We claim that ifp∈[1,∞], then
Mp(µ)⊆Zp∗(µ). (16)
Letx ∈ Mp(µ), and hencex = Pk
i=1ciθiui with Pk
i=1ci|θi|p ≤ 1ifp ∈ [1,∞), and|θi| ≤ 1 fori = 1, . . . , kifp = ∞. Ifp ∈ (1,∞), then it follows from H¨older’s inequality that, for any v ∈Rn, we have
hx, vi= Xk
i=1
ciθihui, vi ≤ Xk
i=1
ci|θi|p
!1p k X
i=1
ci|hui, vi|p∗
!p1∗
≤hZp∗(v).
Ifp= 1, then
hx, vi= Xk
i=1
ciθihui, vi ≤ max
i=1,...,k|hui, vi|=hZ∞(v).
In addition, ifp=∞, then hx, vi=
Xk
i=1
ciθihui, vi ≤ Xk
i=1
ci|hui, vi|=hZ1(v).
Now if p∈ [1,∞), then we deduce from (16) and the reverse Brascamp-Lieb inequality (4) that
V(Zp∗(µ)) ≥ V(Mp(µ)) = 1 Γ(1 + np)
Z
Rn
exp
−kxkpMp
dx
= 2nΓ(1 + 1p)n Γ(1 + np)
Z ∗
Rn
sup
x=Pk i=1ciθiui
Yk
i=1
fi(θi)cidx (17)
≥ 2nΓ(1 + 1p)n Γ(1 + np)
Yk
i=1
Z
R
fi
ci
= 2nΓ(1 + 1p)n
Γ(1 + np) . (18) Finally, ifp=∞, thenfi = 121[−1,1] and
V(Z1(µ))≥V(M∞(µ)) = 2n Z ∗
Rn
sup
x=Pk i=1ciθiui
Yk
i=1
fi(θi)cidx≥2n Yk
i=1
Z
R
fi
ci
= 2n. Equality in (18) leads to equality in the reverse Brascamp-Lieb inequality, and hencek = 2nand u1, . . . , ukform the vertices of a regular crosspolytope inRn.
The main idea in deriving a stability version of (15) and (18) is to establish a stronger version of (8) and (11), respectively, based on the stronger version Lemma 3.1 of (i). In order to apply the estimate of Lemma 3.1, we need some basic bounds on the derivatives of the transportation maps involved. These bounds are proved in Section 4. The technical Sections 5 and 6 also serve as a preparation for the proof of the core statement Proposition 7.2 providing the stabiliy version of (8). The argument for the estimate strenghtening (11) is similar, and is reviewed in Section 8. This finally completes the proof of Theorem 1.4. The stability versions of the reverse isoperimetric inequality in the origin symmetric case (Theorem 1.1 and Theorem1.2) and the strengthening of Theorem 1.4 forp=∞stated in Corollary 1.5 are proved in Section 9.
The methods of this paper are very specific for our particular choice of the functionsfi, and no method is known to the authors that could lead to a stability version of the Brascamp-Lieb inequality (3) or of its reverse form (4) in general. However, the proof of Theorem 1.4 suggests the following conjecture.
Conjecture 2.1 If f is an even probability density function on R with variance 1, g(t) =
√1
2πe−t2/2 is the standard normal distribution, and µ is an even isotropic measure on Sn−1 supported atu1, . . . , uk ∈Sn−1 withµ({ui}) =ci, then
Z
Rn
Yk
i=1
f(hx, uii)cidx ≤ exp (−γmin{1,kf −gk1}α·δWO(µ, νn)α), Z ∗
Rn
sup
x=Pk i=1ciθiui
Yk
i=1
f(θi)cidx ≥ exp (γmin{1,kf −gk1}α·δWO(µ, νn)α), whereγ >0depends onnandα >0is an absolute constant.
3 An auxiliary analytic stability result
To obtain a stability version of Theorem B, we need a stability version of the Brascamp-Lieb inequality and its reverse form in the special cases we use. For this we need some analytic
inequalities such as estimates of the derivatives of the corresponding transportation maps, which will be provided in Section 4. Moreover, we will use the following strengthened form of (i) and a basic algebraic inequality, which were both established in [13, Section 4].
Lemma 3.1 Letk ≥ n+ 1,t1, . . . , tk >0, and letv1, . . . , vk ∈Rn satisfyPk
i=1vi⊗vi = Idn. Then
det Xk
i=1
tivi⊗vi
!
≥θ∗ Yk
i=1
thivi,vii, where
θ∗ = 1 +1 2
X
1≤i1<...<in≤k
det[vi1, . . . , vin]2
√ti1· · ·tin
t0 −1
2
,
t0 =
s X
1≤i1<...<in≤k
ti1· · ·tindet[vi1, . . . , vin]2.
In order to estimateθ∗ from below, we use the following observation from [13].
Lemma 3.2 Ifa, b, x >0, then
(xa−1)2+ (xb−1)2 ≥ (a2−b2)2 2(a2+b2)2.
4 The transportation maps
We note that forp≥1, we have Z
R
e−|t|pdt= 2 p
Z ∞
0
e−ss1p−1ds= 2Γ(1 +1p). (19) Thus forp∈[1,∞], we consider the density functions
̺p(x) =
1
2Γ(1+1p)e−|s|p ifp∈[1,∞),
1
21[−1,1] ifp=∞.
In particular, ̺2 is the Gaussian density function π−1/2e−s2. In addition, we define the trans- portation maps ϕp, ψp : R → R for p ∈ [1,∞), ϕ∞ : (−1,1) → R and ψ∞ : R → (−1,1) by
Z t
−∞
̺p(s)ds =
Z ϕp(t)
−∞
̺2(s)ds, (20)
Z ψp(t)
−∞
̺p(s)ds = Z t
−∞
̺2(s)ds. (21)
Hereϕpandψp are odd and inverses of each other.
In the following, we use that
s−s2 ≤log(1 +s)≤s ifs≥ −12, and the following properties of theΓfunction.
(i)log Γ(t)is strictly convex fort >0;
(ii)Γ(1) = Γ(2) = 1;
(iii)Γ(1 + 2.31 )<Γ(1 + 12) =√ π/2;
(iv)Γ has a unique minimum on (0,∞) at xmin = 1.4616. . . with Γ(xmin) = 0.885603. . ..
In particular, Γ(t) > 0.8856 for t > 0, Γ is strictly decreasing on [0, xmin] and strictly increasing on[1.5,∞).
We deduce from (i)–(iv) that the density functions involved satisfy 1
2e ≤̺p(s)< 1
2·0.8856 forp∈[1,∞]ands∈[0,1]. (22) We note thate/0.8856<3.1, and hence
ϕp(s)∈[0,1)fors ∈[0,3.11 ]. (23) In fact, assuming thatϕp(3.11 )≥1 =ϕp(t),t∈(0,3.11 ], we have
3.1−1 2·0.8856 >
Z t
0
̺p(s)ds= Z 1
0
̺2(s)ds≥ 1 2e, a contradiction. Then, (22) and (5) yield that
1
3.1 < ϕ′p(s), ψp′(s)<3.1 forp∈[1,∞]ands ∈[0,3.11 ]. (24) The following simple estimate will play a crucial role in the proofs of Lemma 4.2 and Lemma 4.3.
Lemma 4.1 Forp∈(1,3)\ {2}andν > 0, letf(t) =νt−ptp−1fort∈[0,1].
(a) Ifp∈(1,2),f(τ)≤0for someτ ∈(0,1]andt∈(0, τ /2], then f(t)<−p(p−1)(2−p)
24−p ·tp−1. (b) Ifp∈(2,3),f(τ)≥0for someτ ∈(0,1]andt∈(0, τ /2], then
f(t)> p(p−1)(p−2) 24−p ·tp−1.
Remark Naturally, the bound could be linear intwith a factor depending onν, but this way the only influence ofνis on the value ofτ. We only use Lemma 4.1 when1.5≤p≤ 2.3andt > c for a positive absolute constantcanyway.
Proof: Let p ∈ (1,2). Since f is convex on [0, τ], τ ≤ 1, f(0) ≤ 0and f(τ) ≤ 0, we have f(2t)≤0fort ∈[0, τ /2]. Taylor’s formula yields that ift∈(0, τ /2], then there existτ1 ∈(0, t) andτ2 ∈(t,2t)such that
0 ≥ 1
2(f(0) +f(2t)) = 1 2
f(t)−f′(t)t+ 1
2f′′(τ1)t2+f(t) +f′(t)t+ 1
2f′′(τ2)t2
= f(t) + 1 2
f′′(τ1) +f′′(τ2) 2 t2,
where 0 < τi < 2t ≤ τ. From f′′(τi) = −p(p−1)(p−2)τip−3 > p(p−1)(2−p)(2t)p−3, i= 1,2, we deduce the estimate
f(t)<−1
2p(p−1)(2−p)(2t)p−3·t2 =−p(p−1)(2−p) 24−p ·tp−1.
Ifp∈(2,3), thenf(t) =νt−ptp−1is concave on[0, τ], and a similar argument yields (b). ✷
Lemma 4.2 Letp∈[1,∞]\ {2}andt ∈(0,18). Then ϕ′′p(t) < −2−p
48 ·t if p∈[1,2), (25)
ϕ′′p(t) > p−2
5 ·t1.3 if p∈(2,3], (26)
ϕ′′p(t) > 0.2·t1.3 if p∈(3,∞]. (27) Proof: For brevity of notation, letϕ = ϕp. We have ϕ(0) = 0 asϕ is odd. Sinceϕ is strictly increasing,ϕ(t)>0ift >0.
Letp∈[1,∞)\ {2}. Fort >0, differentiating (20) yields the formula e−tp
2Γ(1 +1p) = e−ϕ(t)2ϕ′(t) 2Γ(1 +12) , and by differentiating again, we obtain
−pΓ(1 + 12)
Γ(1 + 1p) ·e−tptp−1 =−2e−ϕ(t)2ϕ(t)ϕ′(t)2+e−ϕ(t)2ϕ′′(t).
In particular,
ϕ′(t) = Γ(1 + 12)
Γ(1 + 1p)eϕ(t)2−tp, (28) ϕ′′(t) = (2ϕ(t)ϕ′(t)−ptp−1)ϕ′(t). (29) In the following argument, we use the value
tp = (2/p)p−12 forp∈[1,∞)\ {2}.
The functionp7→ tp is continuously extended top = 2byt2 = e−1/2, and then this function is increasing on[1,∞). In particular,tp ≥1/2forp∈[1,∞).
Moreover, we apply the fact that
for givent ∈(0,1/e),p7→ptp−1is a decreasing function ofp≥1. (30) First, we show that for1≤ p <2andt∈(0,1/4), we haveϕ′′(t)<−248−p·t, which proves (25).
In this case, ϕ′(0) <1by (28), (i), (ii) and (iv). Sinceϕ′ is continuous, there exists a largest sp ∈ (0,∞]such thatϕ′(t) < 1if0 < t < sp. Thus, ift ∈ (0, sp), thenϕ(t) < t, and in turn (29) yields that
ϕ′′(t) = (2ϕ(t)ϕ′(t)−ptp−1)ϕ′(t)<(2t−ptp−1)ϕ′(t).
For1≤p <2andt∈[0, tp], we have2t−ptp−1 ≤0. In particular,ϕ′(t)is monotone decreasing on(0,min{sp, tp}), which in turn implies thatsp ≥tp. We deduce from (24) that
ϕ′′(t)< 2t−ptp−1
3.1 fort∈(0,3.11 ). (31)
Now we distinguish two cases. If1.5≤p <2, then we deduce from (31) and Lemma 4.1 (a) that
ϕ′′(t)<−p(p−1)(2−p)
3.1·24−p ·tp−1 <−
3
4(2−p)
3.1·22.5 ·t <−2−p
24 ·t fort∈(0,14). (32) If1≤ p≤ 1.5, then when estimating the right-hand side of (31) for a givent ∈(0,14), we may assume that p = 1.5 according to (30). In other words, using Lemma 4.1 (a), inequality (32) yields that if1≤p≤1.5andt∈(0,14), then
ϕ′′(t)< 2t−ptp−1
3.1 ≤ 2t−1.5t0.5
3.1 ≤ −2−1.5
24 ·t ≤ −2−p 48 ·t.
Second, if2< p ≤2.3andt∈(0,14), then we show thatϕ′′(t)> p−22 ·t1.3.
In this case,ϕ′(0)>1by (28), (i), (iii) and (iv). Sinceϕ′is continuous, there exists a largest sp ∈(0,∞]such thatϕ′(t)>1if0< t < sp. Thus ift∈(0, sp), thenϕ(t)> t, and in turn (29) yields that
ϕ′′(t) = (2ϕ(t)ϕ′(t)−ptp−1)ϕ′(t)>(2t−ptp−1)ϕ′(t).
Forp > 2andt∈[0, tp], we have2t−ptp−1 ≥0. In particular,ϕ′(t)is monotone increasing on (0,min{sp, tp}), which, in turn, implies thatsp ≥tp. We deduce that
ϕ′′(t)>2t−ptp−1 ift∈(0,12). (33) We deduce from (33) and Lemma 4.1 (b) that
ϕ′′(t)> p(p−1)(p−2)
24−p ·tp−1 > 2(p−2)
22 ·t1.3 = p−2
2 ·t1.3 ift∈(0,14).
Ifp≥2.3andt∈(0,18), thenϕ′′(t)>0.2·t1.3, which completes the proof of (26).
In this case, ϕ′(0) > √
π/2 by (28), (i)–(iv). Since ϕ′ is continuous, there exists largest sp ∈ (0,14]such thatϕ′(t) >√
π/2if0 < t < sp. Thus ift ∈ (0, sp], thenϕ(t) > (√
π/2)·t.
From (30) we see that
2ϕ(t)ϕ′(t)−ptp−1 ≥ π
2 t−ptp−1≥ π
2 t−2.3t1.3 ≥0 for0< t≤sp ≤1/4. Hence (29) yields that
ϕ′′(t) = (2ϕ(t)ϕ′(t)−ptp−1)ϕ′(t)>π
2 t−2.3t1.3
·
√π 2
fort ∈(0, sp]. In particular, we conclude thatsp = 14, and hence Lemma 4.1 (b) yields that ϕ′′(t)> (√
π/2)·2.3·1.3·0.3
21.7 ·t1.3 >0.2·t1.3 fort∈(0,18).
Ifp=∞andt > 0, thenϕ′′(t) > t, which completes the proof of (27). Differentiating (20) we deduce fort∈(−1,1)that
ϕ′(t) = Γ
1 + 1 2
eϕ(t)2 =
√π
2 eϕ(t)2, (34)
ϕ′′(t) = 2ϕ(t)ϕ′(t)2. (35)
Asϕ(t)> 0fort >0, we haveϕ′′(t) ≥ 0by (35), and henceϕ′(t)is monotone increasing for t≥0. Thereforeϕ′(t)≥ϕ′(0) =√
π/2by (34), which, in turn, again by (35) yields that ϕ′′(t)≥2
√ π 2
3
t > t fort∈(0,1).
Thus we have proved all estimates of Lemma 4.2 forϕ′′. ✷
Lemma 4.3 Letp∈[1,∞]\ {2}. Fort∈(0,101), we have ψp′′(t) > 2−p
16 ·t ifp∈[1,2), (36)
